The pointwise densities of the Cantor measure
Let \({\mathcal C}\) be the classical middle-third Cantor set and let \(\mu_{\mathcal C}\) be the Cantor measure. Let \(s = \log 2/ \log 3\) be the Hausdorff dimension of \({\mathcal C}\). For every \(x \in {\mathcal C}\), the authors determine explicitly the values of the upper and lower \(s\)-densities \(\Theta^{* s}(\mu_{\mathcal C}, x)\) and \(\Theta_*^{s} (\mu_{\mathcal C}, x)\) of \(\mu_{\mathcal C}\) at \(x\). In particular, they show that, for \(\mu_{\mathcal C}\)-almost all \(x\), \(\Theta^{* s}(\mu_{\mathcal C}, x)= 2\cdot 4^{-s}\) and \(\Theta_*^{s} (\mu_{\mathcal C}, x) = 4^{-s}\). As a consequence, this latter result shows that the \(s\)-dimensional packing measure of \({\mathcal C}\) is \(4^s\).
scientific article; zbMATH DE number 1542474